The Distribution of the first Eigenvalue Spacing at the Hard Edge of Random Hermitian Matrices.

Nicholas Witte

Abstract: The distribution for the first eigenvalue spacing at the finite, lower endpoint of the spectrum of the Laguerre unitary ensemble of finite rank random matrices is found in terms of the fifth Painlevé system and the solution of its associated linear isomonodromic system. In particular it is characterised by the polynomial solutions to the isomonodromic equations which are also orthogonal with respect to a deformation of the Laguerre weight. In the scaling to the hard edge regime we find an analogous situation where a certain Painlevé III system and its associated linear isomonodromic system characterise the scaled distribution. We undertake extensive analytical studies of this system and use this knowledge to accurately compute the distribution and its moments.

{Joint work with P. Forrester.)